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Plugging in Numbers (Test Prep Algebra!)

 

Sometimes the secret to solving starndardized test algebra problems is just plugging in numbers.  Often the problem will give some parameters for the numbers to plug in (e.g., they must be positive or negative).  Other times, the most efficient way to solve a problem is to plug in the answer choices.  As long as you know the rules of math and algebra, you can plug in numbers and see what your answers come out to!

There are two main types of plugging in questions.  

First, let's deal with the ones that ask for a particular type of number and then want an outcome.  

The basic rule to remember is: If you know what type of number a variable stands for, you can plug in any number that fits the conditions.

You will want to review your math vocabulary so that you know what kinds of numbers are eligible for plugging.

Example: If $x$ represents an even integer, which of the following expressions represent an odd integer?

This question tells you that $x$ is going to be an even integer.  So, the number can be an integer (whole number positive or negative or zero) and that it must be even (a multiple of 2).

Choose a number that meets the condition (start with an easy even integer: 2).

Plug that number into the answer choices to see what answer meets the condition of being an odd integer.

Answer Choices:

a. $x+2$

b. $x-1$

c. $2x$

d. $4x$

e. $5x$

 

Our even integer:

x=2

 

 

 

 

a. $(2)+2=4$  Not odd. 

b. $(2)-1=1$ $\rightarrow\text{Odd integer!}$

c. $(2)\times2=4$  Not odd.

d. $4\times(2)=8$  Not odd. 

e. $5\times(2)=10$  Not odd. 

 

Generally, with these kinds of problems, replace the variable with the simplest variable that fits the condition.  In the above problem, we replaced $x$ with $2$, which is the lowest even integer you can choose.  Only B gives you an odd answer.

For any problem of this type: Choose numbers that are easy to work with.  Some common numbers are 1, 2, and 10. 

1 and 0 are special numbers.  Anything multiplied by 1 is itself.  Anything multiplied by 0 is 0.  SAT loves 1 and 0.

 

Other times, test problems will ask you which of the following answer choices will yield a certain result.  These problems are great!  Just plug in the answer choices and see which one works!

Example: For which of the following values of $y$ is $y^4=(3y)^2$?

This problem might be hard to eyeball.  But, you have five possible answers.  Plug them into check.  While you are plugging, you might find a pattern.  Feel free to jump around the answer choices once you find the pattern.  There's no rule that says you have to plug them in in order.

Answer Choices:

a. $1$

 

b. $2$

 

c. $3$

 

d. $8$

 

e. $9$

 

 

a. $$\eqalign{1^4&=(3(1))^2\\1&\neq9}$$

b. $$\eqalign{2^4&=(3(2))^2\\16&\neq36}$$

c. $$\eqalign{3^4&=(3(3))^2\\81&=81}$$

d. $$\eqalign{8^4&=(3(8))^2\\4096&\neq576}$$

e. $$\eqalign{9^4&=(3(9))^2\\6561&\neq729}$$

 

When you plug these numbers in, you'll see that only 3 (answer c) yields equal answers. 

These questions are test prep problems more than they are math skills -- but they require math skills to execute correctly!  Bottom line: never hesitate to plug in!

Practice Problems:

  • Plugging in Numbers (Test Prep Algebra!)

    1. If the sum of two different integers $x$ and $y$ is 10, what is the greatest possible value of the product of $x$ and $y$?

    a. 9
    b. 16
    c. 24
    d. 25

    2. If the sum of two different integers $m$ and $n$ is 15, what is the greatest possible value of the product of $m$ and $n$?

    a. 49
    b. 50
    c. 56
    d. 60

    3. If the sum of two different integers $r$ and $t$ is 18, what is the greatest possible value of the product of $r$ and $t$?

    a. 36
    b. 77
    c. 80
    d. 81

    4. If the product of two different integers $x$ and $y$ is 24, what is the greatest possible value of the sum of $x$ and $y$?

    a. 10
    b. 11
    c. 13
    d. 25

    5. If the product of two different integers $x$ and $y$ is 60, what is the greatest possible value of the sum of $x$ and $y$?

    a. 12
    b. 16
    c. 32
    d. 61

    6. If the difference of two different positive integers $x$ and $y$ is 3, what is the lowest possible value of the sum of $x$ and $y$?

    a. 4
    b. 5
    c. 6
    d. 7

    7. If $x+3$ is 6 more than $y$, then $x+7$ is how many more than $y$?

    a. 4
    b. 5
    c. 10
    d. 13

    8. If $m+4$ is 5 more than $n$, then $m+10$ is how many more than $n$?

    a. 11
    b. 12
    c. 13
    d. 14

    9. If $x+12$ is 10 more than $y$, then $x+10$ is how many more than $y$?

    a. 8
    b. 10
    c. 12
    d. 14

    10. If $c+2$ is 4 more than $d$, then $c+7$ is how many more than $d$?

    a. 2
    b. 3
    c. 7
    d. 9

Test Prep Practice

  • Algebra: Plugging in Numbers

    1. If $x$ is any negative integer less than $-6$, which of the following must be negative?

    (A) $\dfrac{x^5}{-10}$

    (B) $-2x$

    (C) $x+(-x)$

    (D) $\dfrac{x^2}{x+5}$

    (E) $\dfrac{x}{-2}$

     

     

    2. If $a+b=-20$, and $a>-7$, which of the following must be true?

    (A) $b>-20$

    (B) $b=-13$

    (C) $b<-20$

    (D) $b>-13$

    (E) $b<-13$

     

     

    3. $x(x-3)(x+7)$

    For which of the following values is the value of the expression above negative?

    (A) 3

    (B) -7

    (C) 0

    (D) -9

    (E) 10

     

     

    4. If $mn=9$ and $m$ and $n$ are positive integers, what is a possible value of $4\centerdot m \centerdot m \centerdot \dfrac{m}{n}$?

     

     

     

     

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